Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories

Progress of Theoretical and Experimental Physics, Apr 2015

Supersymmetric gauge theories in five dimensions often exhibit less symmetry than the ultraviolet fixed points from which they flow. The fixed points might have larger flavor symmetry or they might even be secretly 6D theories on $S^{1}$. Here we provide a simple criterion when such symmetry enhancement in the ultraviolet should occur, by a direct study of the fermionic zero modes around one-instanton operators.

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Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories

Prog. Theor. Exp. Phys. Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories Yuji Tachikawa 0 1 0 Institute for the Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 Department of Physics, Faculty of Science, University of Tokyo , Bunkyo-ku, Tokyo 133-0033 , Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supersymmetric gauge theories in five dimensions often exhibit less symmetry than the ultraviolet fixed points from which they flow. The fixed points might have larger flavor symmetry or they might even be secretly 6D theories on S1. Here we provide a simple criterion when such symmetry enhancement in the ultraviolet should occur, by a direct study of the fermionic zero modes around one-instanton operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B10, B14, B16 In four dimensions and in lower dimensions, we often start from a Lagrangian gauge theory in the ultraviolet (UV) and study the behavior of the system in the infrared (IR). In five dimensions (5D), Lagrangian gauge theories are always non-renormalizable, and instead one needs to look for nontrivial ultraviolet superconformal field theories (SCFTs) from which they flow out. With supersymmetry, such a study is indeed possible, as demonstrated in Refs. [1-4], by combining field-theoretical analyses and embedding into string theory. There and in other works, it was found that the UV fixed point can have enhanced symmetry: the instanton number symmetry sometimes enhances to a non-Abelian flavor symmetry, and sometimes enhances to the Kaluza-Klein mode number of a 6D theory on S1. For example, the UV SCFT for N = 1 SU(2) theory with N f 7 flavors has E N f +1 flavor symmetry; with N f = 8 flavors, the UV fixed point is instead a 6D N = (1, 0) theory with E8 symmetry compactified on S1 [5]. For SU(N ) theory, the flavor symmetry enhancement only occurs for some specific choice of the number of the flavors and of the Chern-Simons levels. N = 2 theory with gauge group G in 5D, in contrast, comes from some 6D N = (2, 0) theory compactified on S1, as originally found in the context of string duality [6,7]. These results were soon extended to include more models, using webs of five-branes, in, e.g., Refs. [8-10]. More recently, various sophisticated techniques such as supersymmetric localizations, Nekrasov partition functions, and refined topological strings have been applied to the analysis of these 5D systems. The symmetry enhancement of the models mentioned above has been successfully confirmed by these methods, and even more diverse models are being explored; see, e.g., Refs. [11-27]. 1. Introduction These results are impressive, but the techniques used are rather unwieldy.1 In this paper, we describe a simpler method to identify the symmetry enhancement of a given 5D gauge theory, assuming that it has an ultraviolet completion either in 5D or 6D. Although heuristic, this method tells us what the enhanced flavor symmetry will be and whether the ultraviolet completion is a 5D SCFT or a 6D SCFT on S1. We do this by identifying the supermultiplet of broken symmetry currents2 by studying instanton operators that introduce non-zero instanton number on a small S4 surrounding a point.3 When the instanton number is one, the structure of the moduli space and the zero modes is particularly simple, allowing us to find the broken symmetry currents rather directly. The spirit of the analysis will be close to the analysis of monopole operators of 3D supersymmetric gauge theories in Ref. [30], except that we need to identify an operator in the IR that would come from a UV current operator, rather than vice versa. The rest of the paper is organized as follows. In Sect. 2, we collect the basic facts on the supersymmetry and on instantons on S4 that we use. In Sect. 3, we analyze N = 1 SU(2) gauge theories with N f 8 flavors and N = 2 SU(2) theory. The effect of the discrete theta angle will also be discussed. In Sect. 4, we extend the discussion to N = 1 SU(N ) gauge theories with fundamentals and ChernSimons terms, and to N = 2 SU(N ) theory. We use the results that will be obtained up to this point in Sect. 5 to study symmetry enhancement in quiver gauge theories made of SU gauge groups and bifundamentals. We assume that the effective number of flavors of each SU(N ) node is 2N and that the ChernSimons levels are all zero. We conclude with a discussion in Sect. 6. 2. Preliminaries 2.1. Supermultiplet of broken currents with scaling dimensions 3, 3.5, 4, 4, respectively. Here a is the adjoint index of the flavor symmetry, i = 1, 2 is the index of Sp(1)R , and is the spinor index of SO(5); the symplectic Majorana condition is imposed on ia. 1 One of the trickiest aspects is the need to remove spurious contributions from the so-called U(1) parts in the instanton counting method and from the parallel legs in the toric diagram in the topological string method. 2 The importance and the special property of the current supermultiplets in 5D SCFTs were emphasized in Ref. [13]. The analysis presented here is strongly influenced by that paper. 3 Recently, a paper [28] appeared in which instanton operators of 5D N = 2 gauge theories were also studied. There the emphasis was on the multi-point functions of instanton operators. In this paper we just consider a single instanton operator, and the multiplet it forms under the flavor symmetry and the supersymmetry. In another recent paper [29], a different type of instanton operator was studied, where an external Sp(1)R background was introduced on S4 to have manifest supersymmetry at the classical level. Here we do not introduce such additional backgrounds. 4 For the details of the superconformal multiplets in 5D and in 6D, see, e.g., Ref. [31] and references therein. The mass deformation is done by adding L = ha Ma to the Lagrangian, thus breaking some of the flavor symmetry: where (i j) is an Sp(1)R triplet scalar (i j), i is a symplectic Majorana Sp(1)R doublet fermion i, and J is a vector. The component M in (2.1) is a divergence of J, and therefore is a descendant. Suppose instead that the 5D theory is obtained by putting an N = (1, 0) 6D SCFT on S1, possibly with a nontrivial holonomy for the flavor symmetry; we consider N = (2, 0) theories as special cases of N = (1, 0) theories. The conserved current supermultiplet in 6D contains where A is the 6D vector index and we have omitted the adjoint index for brevity. On S1, we have KaluzaKlein (KK) modes J(n) and M(n) := J6(n) where n is the KK mode number. Then the 6D conservation gives 2.2. Instanton operators Suppose now that we are given a 5D gauge theory, and further assume that it is a mass deformation of a 5D SCFT or a 6D SCFT on S1. Pick a point p in R5 and insert there a current operator that is broken by the mass deformation in the former case, and a current operator with non-zero KK mode number in the latter case. Surround p by an S4, on which we have a certain state. Let the size of S4 be sufficiently larger than the characteristic length scale of the system set by the mass deformation or the inverse radius of S1. Then the state on S4 can be analyzed using the gauge theory. When the instanton number of the gauge configuration on S4 is non-zero, we call the original operator inserted at p an instanton operator. It should be possible to study the supermultiplet structure of instanton operators in detail using supersymmetric 5D Lagrangians on S4 times R or S1, using the results in, e.g., Refs. [11,3234]. Here we only provide a rather impressionistic analysis of one-instanton operators, i.e., the instanton operators when the instanton number on S4 is one. In the rest of this section we gather known facts on one-instanton moduli spaces and fermion zero modes. We will be brief; a comprehensive account of SU(N ) instantons can be found in, e.g., Ref. [35]. For instantons of general gauge groups, see, e.g., Refs. [36,37]. When the gauge group is SU(2). Any instanton configuration on S4 can be obtained by conformal transformations from one on a flat R4. When the gauge group is SU(2), the one-instanton moduli space on R4 has the form where R4 parametrizes the position, R>0 the size, and S3/Z2 the gauge rotation at infinity. When the instanton is mapped to S4, the first two factors R4 and R>0 combine to form the ball B5 with a standard hyperbolic metric.5 The asymptotic infinity of R4 is mapped to a point on S4, and therefore the gauge symmetry there should really be gauged. Therefore we lose the last factor S3/Z2, but we should remember that the gauge group SU(2) is broken to Z2. A point in B5 very close to a point x in S4 describes an almost point-like instanton localized at x . A point at the center of B5 corresponds to the largest possible instanton configuration, which is in fact SO(5) invariant. This invariant configuration can be identified with the positive-chirality spinor bundle on the round S4. This configuration is also known as Yangs monopole [42]. A Weyl fermion of the correct chirality in the doublet of the SU(2) gauge group has just one zero mode on a one-instanton configuration. On S4, this is a singlet of SO(5) rotational symmetry. A Weyl fermion of the correct chirality in the adjoint of the SU(2) gauge group has four zero modes. Two are obtained by applying supertranslations and the other two by applying special superconformal transformations to the original bosonic configuration. When conformally mapped to S4, these four modes transform in the spinor representation of SO(5). They are exactly the modes obtained by applying the 5D supersymmetry to the bosonic instanton configuration. In this sense the instanton configuration breaks all the supersymmetry classically, but this does not mean that the instanton operator is in a generic, long multiplet of supersymmetry, as we will soon see. When the gauge group is general. When the gauge group is a general simple group G, any one-instanton configuration is obtained by embedding an SU(2) one-instanton configuration by a homomorphism : SU(2) G determined by a long root. Therefore, the one-instanton moduli space on R4 is of the form where H is the part of G that is unbroken by the SU(2) embedded by . A further quotient G/(SU(2) H ) is known as the Wolf space of type G. The form of H is well known; here we only note that for G = SU(N ) we have H = U(1) SU(N 2). The conformal transformation to S4 again combines R4 and R>0 to the ball B5, and we lose G/H as before. The remaining effect is that we have H as the unbroken gauge symmetry. Analyzing the fermionic zero modes of an arbitrary representation R of G around this configuration is not any harder than for SU(2), since the actual gauge configuration is still essentially that of SU(2). We only have to decompose R under SU(2) H , and to utilize our knowledge for SU(2). 5 The hyperbolic ball B5 is also known as the Euclidean AdS5. This fact was used effectively in a series of early works relating AdS/CFT and instantons; see, e.g., Refs. [3841]. 3. SU(2) 3.1. Pure N = 1 theory After these preparations, let us first consider one-instanton operators of pure N = 1 SU(2) gauge theory. As recalled in the previous section, the one-instanton moduli space on S4 is just the ball B5. We consider a state corresponding to the lowest SO(5)-invariant wavefunction. Now we need to take fermionic zero modes into account. The gaugino is a spinor field in five dimensions, which gives two Weyl fermions with the correct chirality in the adjoint of the SU(2) gauge group and in the doublet of Sp(1)R when restricted on S4. As recalled in the previous section, the zero modes are in the doublet of Sp(1)R and in the spinor of SO(5) rotational symmetry. Let us denote them by i, where i = 1, 2 is for Sp(1)R and = 1, 2, 3, 4 is for SO(5). The symplectic Majorana condition in 5D means that we need to quantize these zero modes into gamma matrices satisfying The states on which these zero modes act, then, can be found by decomposing the Dirac spinor representation of SO(8) in terms of its subgroup Sp(1)R SO(5) such that the vector representation of SO(8) becomes the doublet times the quartet. We find the following sixteen states: which form exactly the broken current supermultiplet (2.3) recalled in the last section. We put the plus signs as superscripts to remind us that they are one-instanton operators. Let us assume that this gauge theory is a mass deformation of a UV 5D SCFT. Then the UV SCFT should simultaneously have both the multiplet that contains J+ in the gauge theory and the multiplet that becomes the instanton number current Now, J+ has charge 1 under J0, because J+ is a one-instanton operator. Therefore, they should form an SU(2) flavor symmetry current. This conclusion agrees with the original stringy analysis [1]. Effect of the discrete theta angle. In the analysis so far, we have neglected the effect of the discrete theta angle of the SU(2) theory. The theta angle is associated with 4(SU(2)) = Z2, and therefore only takes the values = 0 or . On a five-manifold of the form M S1 such that the SU(2) configuration on M has instanton number 1 and there is a nontrivial Z2 holonomy around S1, the theta angle = assigns an additional sign factor 1 in the path integral.6 On R4 S1, this has an effect that makes the wavefunctions on the one-instanton moduli space sections of a nontrivial line bundle with holonomy 1 on S3/Z2. In our setup, the supermultiplet (3.2) is kept when = 0, but is projected out when = . Therefore, there is an enhancement of the instanton number symmetry to SU(2) when = 0, but we see no enhancement when = . This effect of the discrete theta angle matches what was found originally in Ref. [2] using stringy analysis. A caveat here is that, in our crude analysis, we can only say that the states considered so far do not give any broken current supermultiplet. It is logically possible that exciting non-zero modes in the one-instanton sector or considering operators with instanton number 2 or larger gives rise to a 6 For a derivation, see, e.g., Appendix A of Ref. [43]. broken current supermultiplet, enhancing the instanton number symmetry to SU(2) even with = . A careful study on this point is definitely worthwhile, but is outside the scope of this paper. The same caveat is also applicable to the rest of the article, but we will not repeat it. 3.2. With fundamental flavors Next, let us consider N = 1 SU(2) theory with N f flavors in the doublet. Stated differently, we add 2N f half-hypermultiplets in the doublet of SU(2). At the classical Lagrangian level, they transform under SO(2N f ) symmetry. In a one-instanton background, they give 2N f fermionic zero modes, which need to be quantized as gamma matrices: { where a = 1, . . . , 2N f is the index of the vector representation of SO(2N f ). They act on the Dirac spinor representation S+ S of SO(2N f ). Tensoring with the result (3.2) of the quantization of the adjoint zero modes, we find that the one-instanton operator is a broken current supermultiplet in the Dirac spinor of SO(2N f ). Now we need to take the unbroken Z2 gauge symmetry into account. The generator of this Z2 symmetry acts by 1 on the doublet representation, and therefore by 1 on the gamma matrices in (3.4). Therefore, this Z2 acts as the chirality operator on S+ S. Therefore, depending on the value of the discrete theta angle being 0 or , we keep only one-instanton operators in S+ or S. These two choices are related by the parity operation of flavor O(2N f ) and are therefore equivalent. In conclusion, we found the following current multiplets: the conserved SO(2N f ) currents J[ab], the conserved instanton number current J0, the broken currents coming from one-instanton operators J+,A where A is the chiral spinor index of SO(2N f ). Take 1 N f 7. If we assume that the gauge theory is a mass deformation of a UV fixed point, we see that the currents listed above need to combine to give the flavor symmetry E N f +1. This can be seen by attaching an additional node, representing a Cartan element for J0, to the node of the Dynkin diagram of SO(2N f ) that gives the chiral spinor representation. For example, we have when N f = 7, where the black node is for the instanton number current.7 This enhancement pattern agrees with what was found originally in Ref. [1]. Let us boldly take N f = 8. We now have the combined Dynkin diagram which is known as E9 or E8. If we assume that the gauge theory is an outcome of a massive deformation of some UV completion, the UV completion needs to have an affine E8 flavor symmetry as 7 Strictly speaking, this procedure is not unique when a theory with very small N f is considered alone. As a simpler example of this issue, suppose we know that we have an SU(2) current, an instanton number current, and a broken current coming from one-instanton operators in the doublet of SU(2). Then we have two choices, either an SU(3) corresponding to or an Sp(2) corresponding to . They can only be distinguished by studying the two-instanton operators. In our case, however, we can just study the N f = 7 case and then apply the mass deformation to make N f smaller, to conclude that the flavor symmetry is always EN f +1. a 5D theory. Stated differently, this means that the UV completion needs to be a 6D theory with E8 flavor symmetry. This conclusion matches what was found in Ref. [5]. We see a problem when N f > 8: the combined Dynkin diagram defines a hyperbolic KacMoody algebra, whose number of roots grows exponentially. It is therefore unlikely that there is a UV completion when N f > 8. Again, this matches the outcome of a different analysis in Ref. [1]. Before proceeding, we note here that it is well known that the instanton particles in this gauge theory give rise to spinors of SO(2N f ) flavor symmetry. The only slightly new point in this section is that, when studied in the context of instanton operators, they are indeed part of the broken current multiplets given by (3.2). 3.3. N = 2 theory As a final example of SU(2) gauge theory, let us consider N = 2 gauge theory. On the one-instanton background on S4, we have four Weyl fermions in the adjoint of SU(2), and the zero modes can be denoted by i where i = 1, 2, 3, 4 is now for Sp(2)R and = 1, 2, 3, 4 is for the spacetime SO(5). Again, the symplectic Majorana condition in 5D means that they become gamma matrices with the commutation relation These gamma matrices act on the following one-instanton states on S4: where a = 1, 2, 3, 4, 5 is the vector index for Sp(2)R = SO(5)R. The gamma-tracelessness condition on +, Q+ and the tracelessness condition on +, T + need to be imposed. When the discrete theta angle is zero, these operators are all kept, and the structure of the operators is exactly that of KK modes of the 6D N = 2 energymomentum supermultiplet. For example, the currents J[+ab] in the adjoint of Sp(2)R suggest that the Sp(2)R symmetry of the 5D gauge theory enhances to the affine Sp(2)R, and the symmetric traceless T(+) is the KK mode of the 6D energy momentum tensor. This is as it should be, since the S1 compactification of N = (2, 0) theory of type SU(2) on S1 is described by N = 2 SU(2) gauge theory in 5D. The relation between the 5D N = 2 theory and the 6D N = (2, 0) theory on S1 has been extensively studied; see, e.g., Refs. [44,45]. When the discrete theta angle is , these operators are all projected out. In this case too, the gauge theory is the IR description of N = (2, 0) theory of type SU(3) on S1 with an outer-automorphism Z2 twist around it [43]. We expect operators with the same structure to arise in a sector with higher instanton number, but to check this is outside the scope of this paper. 4. SU(N ) 4.1. Pure N = 1 theory Now let us move on to SU(N ) gauge theories. Our first example is the pure SU(N ) theory. As recalled in Sect. 2, in one-instanton configurations on S4, the gauge fields take values in SU(2) SU(N ). The unbroken subgroup is U(1) SU(N 2). We take the generator of the U(1) part to be diag(N 2, N 2, 2, 2, . . . , 2). In this normalization, when the ChernSimons level is , the one-instanton configuration has the U(1) charge (N 2). An adjoint Weyl fermion of SU(N ) on this background decomposes into an SU(2) adjoint Weyl fermion, N 2 Weyl fermions in the doublet of SU(2) in the fundamental of SU(N 2) and with U(1) charge N , and Weyl fermions that are neutral. The gaugino in 5D gives two adjoint Weyl fermions of SU(N ) on S4. The SU(2) adjoint part gives the same broken current supermultiplet (3.2), and we need to take additional 2 (N 2) zero modes coming from the SU(2) doublet into account. By quantizing them, we have fermionic creation operators Bia where i = 1, 2 is now for Sp(1)R and a = 1, . . . , (N 2) is for SU(N 2). The U(1) charge of Bia is N . The SU(N 2) invariant states are then a1aN2 Bi1a1 Bi2a2 BiN2aN2 |0 , (Bia)2(N 2) |0 , with U(1) charge (N 2)N , 0, +(N 2)N , respectively. When is neither 0 nor N , all states are projected out, due to the non-zero U(1) gauge charge. When is N , one singlet state is kept, and we have a broken current supermultiplet (3.2). Therefore we expect the enhancement of the instanton number symmetry to SU(2). This enhancement was recently discussed in Ref. [19]. When is 0, the one-instanton operators are the tensor product of the broken current supermultiplet (3.2) times a1aN2 Bi1a1 Bi2a2 BiN2aN2 |0 , which transforms in the N 1-dimensional irreducible representation of Sp(1)R. This is a short supermultiplet, but does not correspond to a broken flavor symmetry. 4.2. With fundamental flavors Our next example is SU(N ) theory with N f hypermultiplets in the fundamental representation. On one-instanton configurations on S4, each flavor decomposes into a pair of SU(2) doublets and a number of neutrals; they all have U(1) charge N 2. Therefore, we have additional fermionic creation operators Cs , s = 1, . . . , N f , of U(1) charge N 2. They act on the states of the form for k = 0, . . . , N f , with U(1) charge (N 2)(k N f /2). Tensoring (3.2), (4.2), and (4.3) and imposing the U(1) gauge neutrality condition, we see that a broken symmetry supermultiplet survives when we have Now, in Ref. [4] it was shown that we need || N N f /2 to have a 5D UV SCFT behind the gauge theory. Therefore | N | N f /2. We also trivially have |k N f /2| N f /2. Therefore, the equality (4.4) can only be satisfied when = N N f /2. When = N N f /2, the surviving broken current supermultiplet comes from k = 0 or k = N f in (4.3). They have instanton number one and baryonic charge N f /2. Therefore, when = N N f /2, one U(1) enhances to SU(2), and, when = N N f /2, another U(1) enhances to SU(2). When = N N f /2 = 0, the combination I B/N of the instanton charge I and the baryonic charge B both enhance to SU(2), making the UV flavor symmetry SU(N f ) SU(2)+ SU(2). This enhancement pattern was found in Refs. [19,27]. 4.3. N = 2 theory Let us next consider N = 2 SU(N ) theory. The ChernSimons level is automatically zero. Again, the adjoint Weyl fermions of SU(N ) decompose into those that are adjoint of SU(2) and those that are doublets of SU(2). The zero modes of the former generate the states (3.8). The zero modes of the latter give us fermionic creation operators Bia, where i = 1, 2, 3, 4 is for Sp(2)R and a = 1, . . . , (N 2) is for gauge SU(N 2). The U(1) SU(N 2) neutral states then have the form a1aN2 Bi1a1 Bi2a2 BiN2aN2 b1bN2 B j1b1 B j2b2 B jN2bN2 |0 = Bi1 1 B j1 1 Bi2 2 B j2 2 BiN2 N 2 B jN2 N 2 |0 . We want to decompose them under the action of Sp(4)R. Let us think that SU(4) acts on the indices i and j . The indices in and jn are antisymmetrized. Combined, [in jn] is a vector of SO(6) SU(4). The indices are symmetrized under the combined exchange of [in jn] and [im jm ]. Therefore, the states (4.5) transform under the (N 2)nd symmetric power of the vector of SO(6). Decomposing it under Sp(4)R SO(5)R SO(6), we see that the states (4.5) are in the representation V0 V1 VN 2, where Vk is the kth symmetric traceless representation of SO(5)R. This structure is precisely what we would expect for the KK modes of N = (2, 0) theory of type SU(N ) put on S1. In general, N = (2, 0) theory of type G has short multiplets containing a spacetime symmetric traceless tensor that is in Vn2 of SO(5)R, for each n that gives a generator of invariant polynomials of G. For G = SU(N ), n runs from 2 to N , thus giving (4.6) tensored with (3.8). It would be interesting to perform similar computations for N = 2 theories for other G. For example, when G = E6, we should have V0 V3 V4 V6 V7 V10, since the generators of invariant polynomials of E6 have degrees 2, 5, 6, 8, 9, and 12. When G is non-simply laced, the UV completion is N = (2, 0) theory of some simply laced type, with an outer-automorphism twist around S1. This again predicts which Vn should appear among the one-instanton operators. 5. Quivers In this section, we study the symmetry enhancement in the quiver gauge theory with SU gauge groups and bifundamental hypermultiplets. In this note we do not aim at comprehensiveness; instead we only treat the case where the effective number of flavors at each node SU(Ni ) is 2Ni and the ChernSimons levels are all zero. We use the by-now standard notation where stands for an SU(N1) flavor symmetry node and an SU(N2) gauge symmetry node connected by a bifundamental hypermultiplet, etc. We also use a special convention that two flavors of SU(1), , stand for . The rationale behind this convention will be explained later in Sect. 5.3. 5.1. SU(2)2 theory We denote the gauge groups as SU(2)1 SU(2)2. The hypermultiplets Q0, Q1, Q2 have flavor symmetries SO(4)F0, SU(2)F1, SO(4)F2, respectively. The gauge group SU(2)1 effectively has N f = 4 flavors, and thus it has E4+1 = SO(10) flavor symmetry when SU(2)2 is not gauged. After gauging, the remaining flavor symmetry is the commutant of SU(2)2, which is SU(4) SU(2). We can summarize this enhancement pattern as where the two white nodes on the left are SO(4)F0, the white node on the right is SU(2)F , and the black node is the contribution from the instanton operator of SU(2)1. The same argument can be applied to the SU(2)2 side, and we conclude that the full flavor symmetry A B1 C CB1 BA2 AB2 , (5.4) where each symbol stands for a 2 2 block. The blocks A, A , and A are three SU(2) flavor symmetries that can be seen in the Lagrangian; Bi comes from one-instanton operators of the SU(2)i gauge group; and C comes from instanton operators that have instanton number one for both gauge groups SU(2)1,2. Let us call these last ones (1, 1)-instanton operators. They transform as chiral spinors under SO(4)F0,F2 and are neutral under SU(2)F . Let us try to study the (1, 1)-instanton operators directly. The zero modes of gauginos of SU(2)1,2 give two copies of the broken current multiplets (3.2); those of hypermultiplets Q0 and Q2 give the spinors of SO(4)F0 and SO(4)F2. Finally, the hypermultiplet Q1 couples to one-instanton configurations of both SU(2)1,2. Therefore this is effectively a triplet coupled to an SU(2) one-instanton configuration. It is also a doublet of SU(2)F . Therefore they give rise to the states where a, b = 1, 2 is now the index of the doublet of SU(2)F . We therefore need to take the tensor product of two copies of the broken current multiplet (3.2), the spinors of SO(4)F0,F2, and the multiplet (5.5). We then need to impose the Z2 projections for SU(2)1,2. At present we do not know enough about the behavior of the tensor product of the supersymmetry multiplets in 5D. Instead, we learn the following by using the knowledge of the flavor symmetry properties of (1, 1)-instanton operators deduced from the block decomposition (5.4): the tensor product of two copies of the broken current multiplet (3.2) and the multiplet (5.5) contains again a unique broken current multiplet. Furthermore, it is SU(2)F neutral, and therefore it comes from the factor J+ in (5.5). 5.2. SU(N1) SU(N2) theory We assume N1 > 2, N2 > 2, N0 + N2 = 2N1, and N1 + N3 = 2N2. We also set both the Chern Simons levels to zero. Let us denote by U(1)B1 and U(1)B2 the baryonic flavor symmetries that assign charge 1 to a field in the fundamental of SU(N1) and SU(N2), respectively. Let us also denote by U(1)I1 and U(1)I2 the instanton number charge of SU(N1) and SU(N2), respectively. We already saw that the combinations I1 := I1 B1/N1 and I2 := I2 B2/N2 are each enhanced to an SU(2), giving SU(2)4 flavor symmetry. Let us show that I1+ and I2+ combine to form an SU(3)+ and similarly that I1 and I2 combine to form an SU(3). To see this, we need to analyze (1, 1)-instanton operators in this theory. The gauge group SU(Ni ) is broken to U(1)i SU(Ni 2). The gaugino zero modes can be analyzed as before. The hypermultiplets Q0 and Q2 give doublets of SU(2) one-instanton configuration; the hypermultiplet Q1 similarly gives a lot of doublets and just one triplet of SU(2) one-instanton configuration. Then, we need to find states that are neutral under the unbroken gauge group from the tensor product of the following contributions: 1. From fields that are doublets of the SU(2) one-instanton configuration, we have 1a. contributions (4.2) from gauginos of SU(N1) and SU(N2), 1b. and contributions (4.3) from Q0, Q1, and Q2. 2. From fields that are triplets of the SU(2) one-instanton configuration, we have 2a. a contribution (5.5) from Q1, 2b. and two copies of the broken current multiplet (3.2) from SU(N1,2). From the contributions 1, we find two states that are neutral under the unbroken gauge group U(1)1 SU(N1 2) U(1)2 SU(N2 2), by tensoring the ground state or the top state of (4.3) from the contributions 1a by the ground state or the top state of (4.3) from the contributions 1b. From the contributions 2a, we note that the only U(1)1- and U(1)2-neutral state is the component J in (5.5). Tensoring with the contributions 2b, we find a broken current multiplet, as we found at the end of the last subsection. In total, we find at least two gauge-invariant broken current multiplets. The charges under the Lagrangian flavor symmetries can be easily found: both are neutral under SU(N0) and SU(N3), and the charges under U(1)Q0, U(1)Q1, U(1)Q2 are Therefore we have found two broken current multiplets with charges under (I1+, I2+; I1, I2) given by (1, 1; 0, 0) and (0, 0; 1, 1), respectively. We already know that one-instanton operators of SU(N1) give broken current multiplets with charges (1, 0; 0, 0) and (0, 0; 1, 0), and similarly those of SU(N2) give multiplets with charges (0, 1; 0, 0) and (0, 0; 0, 1). Therefore we see that the instanton number currents I1+, I2+ combine to form SU(3)+, and the currents I1 and I2 combine to form SU(3). The total flavor symmetry is therefore at least SU(3)+ SU(3) SU(N0) SU(N3) U(1). The last U(1) is absent when N0 or N3 is zero. 5.3. Some special two-node quivers We need to analyze separately the cases when one of the gauge groups is SU(2) or SU(1). As already mentioned, we use the convention where two flavors of SU(1) We also apply the same convention where the SU(2) on the left is gauged. From a purely 5D field theory point of view, this is really just a convention, but it is useful because SU(1) with two flavors shows an enhanced symmetry of SU(2) SU(2) SU(2), just as a special case of SU(N ) with 2N flavors with the symmetry enhancement SU(2) SU(2) SU(2N ). From a string/M theory point of view, when SU(1) with two flavors is engineered, say, in the brane web construction, one in fact finds additional hypermultiplets coming from point-like SU(1) instantons that naturally give rise to the setup (5.10). This is another rationale for our convention. Now, let us couple this SU(1) with two flavors to an SU(2) gauge group to form a two-node quiver: Using our convention, this is just SU(2) with five flavors that show an enhancement to E6. This contains SU(3)+ SU(3) SU(3), showing the general pattern that we found in (5.8). We already treated and saw that the symmetry is SU(2) SU(6) SU(2). As SU(6) SU(3)+ SU(3) U(1), it again shows the general pattern (5.8). Finally, let us consider The SU(2) theory before coupling to SU(4) has an enhanced symmetry SO(10). After coupling to SU(4) the remaining part is SU(2)+ SU(2), and they are enhanced by the dynamical SU(4) to SU(3)+ SU(3), again following the general pattern (5.8). 5.4. Multi-node quivers After our preparation on the two-node quivers, it is easy to analyze general multi-node quivers, again with the restriction that each SU(N ) node has effectively 2N flavors and zero ChernSimons terms. Consider as an example the quiver Each SU(Ni ) node with Ni = 4, 3, 2, 1 shows an enhancement of the linear combination of the instanton current and the baryonic current, Ii = Ii Bi /Ni to SU(2)i. For each neighboring pair of nodes SU(Ni ) SU(N j ), SU(2)i and SU(2) j enhance to form SU(3). Therefore, in total, we should have SU(5)+ SU(5) from the enhancement of the instanton number symmetry and the baryonic symmetry. Combined with the original flavor symmetry SU(5) of the leftmost node, we have SU(5)+ SU(5) SU(5) as the enhanced symmetry. We can easily generalize this analysis to an analogous linear quiver with the gauge group SU(N 1) SU(N 2) SU(2) SU(1), with bifundamentals between the neighboring gauge nodes and additional N fundamentals for SU(N 1); we see the symmetry SU(N )+ SU(N ) SU(N ). The 5D SCFT is called the 5D TN theory, and this linear quiver presentation was recently studied in Refs. [17,20,25,26]. As another example, consider the following quiver: We have gauge groups SU(Ni ) with i = 1, 2, . . . , 9, with Ni = ki m. Let us define Ii = Ii Bi /Ni . Then, applying exactly the same argument as above, we see that the currents Ii+ combine to form (E8)+ and the currents Ii enhance to (E8). The total flavor symmetry is not quite their product, however. The Cartan generator corresponding to a pure KK momentum of (E8)+ is Bi does not act on the hypermultiplets and therefore is trivial. Thus we have meaning that the total flavor symmetry is ki Ii+ = 1 ki Ii + m ki Ii+ = ki Ii, showing that the possible UV completion of this gauge theory is a 6D SCFT with flavor symmetry E8 E8 on S1. This is as expected: m M5-branes on the ALE singularity of type E8 gives a 6D N = (1, 0) SCFT in the infrared, with E8 E8 flavor symmetry. Compactifying it on S1 and reducing it to type IIA, we have m D4-branes probing the ALE singularity of type E8. Using the standard technique [46], we find the quiver theory given above. The general statement is now clear. Take a 5D quiver gauge theory, with each SU(N ) gauge node having effectively 2N flavors. If the quiver is a finite simply laced Dynkin diagram of type G, the instanton number currents enhance to G G; if the quiver is an affine simply laced Dynkin diagram of type G, the instanton number currents enhance to G G. 6. Conclusions In this paper we have analyzed the one-instanton operators of 5D gauge theories with SU(N ) gauge groups with hypermultiplets in the fundamental, adjoint, or bifundamental representations. We saw that a simple exercise in the treatment of fermionic zero modes gives rise to the expected patterns of symmetry enhancements. There are many areas to be further explored. One is to extend our analysis to include SU(N ) gauge theories with other matter representations, such as antisymmetric or symmetric two-index tensor representations, and to consider other gauge groups, both classical and exceptional. There should not be any essential difficulty in performing this generalization, since a one-instanton configuration in any group G is always just an SU(2) one-instanton configuration embedded into G. Our analysis of the SU quiver theory is by no means exhaustive, and it would be interesting to consider more general cases. It might be interesting to study instanton operators with higher instanton numbers. This will be significantly harder, however, since the instanton moduli space is much more complicated. Presumably, we will need to use the localization etc. to analyze it, and the method would become equivalent to what has already been done in the literature in the study of the superconformal index of the 5D SCFTs. Another direction is to study in more detail the structure of the supermultiplets formed by operators in non-conformal 5D supersymmetric theories. In this paper we relied on some heuristics based on the known supermultiplet structures of superconformal theories. The gauge theories in the infrared are, however, non-conformal, and we should analyze them as they deserve. For example, in our analysis of SU(N1) SU(N2) theory, we could not directly analyze the tensor product decomposition of the two copies of (3.2) and the contribution (5.5); instead we needed to import the knowledge gained by the analysis of the special case SU(2)2. This is not an ideal situation. With a proper understanding of the supermultiplet structures of operators in non-conformal theories, we would be able to analyze this tensor product directly. We also assumed throughout this paper that we only have to consider fermionic zero modes around the one-instanton configuration, and that the states with excited non-zero modes do not give broken current supermultiplets. This is at least plausible, since non-zero modes would likely produce descendant operators, but this is not at all a rigorous argument. This needs to be better investigated. Finally, we assumed in this paper that the gauge theory that we analyze is a mass deformation of a UV fixed point, either a 5D one or a 6D one compactified on S1, and then studied what would be the enhanced symmetry in the ultraviolet. It would be desirable to understand the criterion to tell which 5D gauge theory has a UV completion. The author would like to come back to these questions in the future, but he will not have time in the next few months due to various duties in the university. He hopes that some of the readers get interested and make great progress in the meantime. Y.T. thanks the Yukawa Institute at Kyoto University, for inviting him to give a series of lectures in February 2015. He, without much thought, promised to give an introduction to supersymmetric field theories in five and six dimensions. It was after much thought about how to organize the content that he came up with the arguments presented in this manuscript. 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Yuji Tachikawa. Instanton operators and symmetry enhancement in 5D supersymmetric gauge theories, Progress of Theoretical and Experimental Physics, 2015, DOI: 10.1093/ptep/ptv040